The animal best off is outside with the dogs (7)
I believe the answer is:
bassets
'dogs' is the definition.
(basset is a kind of dog)
'the animal best off is outside' is the wordplay.
'the animal' becomes 'ass' (term for a donkey).
'off' indicates an anagram.
'is outside' means one lot of letters goes inside another.
'best' anagrammed gives 'bets'.
'ass' going within 'bets' is 'BASSETS'.
'with the' is the link.
(Other definitions for bassets that I've seen before include "Hounds with short legs and long ears" , "Dogs" .)